|
|
A349274
|
|
Decimal expansion of the arc length of the logarithmic spiral related to a golden triangle with a unit base length.
|
|
1
|
|
|
4, 9, 2, 5, 1, 7, 5, 4, 3, 9, 4, 7, 8, 0, 5, 8, 8, 3, 7, 1, 9, 3, 1, 7, 6, 0, 4, 6, 8, 1, 3, 5, 6, 5, 6, 7, 4, 2, 2, 4, 5, 7, 0, 1, 9, 2, 8, 4, 0, 6, 9, 5, 2, 2, 2, 1, 5, 2, 0, 5, 8, 0, 2, 2, 0, 9, 4, 6, 3, 2, 5, 2, 4, 4, 3, 4, 6, 5, 2, 2, 6, 3, 3, 9, 0, 1, 5, 4, 3, 1, 5, 2, 0, 7, 8, 4, 4, 6, 8, 9, 5, 8, 3, 4, 8
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The spiral starts at the triangle's apex (the vertex opposite to the base) and converges to the pole (the accumulation point).
|
|
REFERENCES
|
H. E. Huntley, The Divine Proportion: A Study in Mathematical Beauty, New York: Dover Publications Inc., 1970, pp. 170-176.
Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number, New York: Broadway Books, 2002, p. 119.
|
|
LINKS
|
|
|
FORMULA
|
Equals sqrt((17 + 7*sqrt(5))/22) * sqrt(1 + b^2)/b, where b = 5*log(phi)/(3*Pi), and phi = (1+sqrt(5))/2 is the golden ratio (A001622).
|
|
EXAMPLE
|
4.92517543947805883719317604681356567422457019284069...
|
|
MATHEMATICA
|
b = 5 * Log[GoldenRatio]/(3*Pi); RealDigits[Sqrt[(17 + 7*Sqrt[5])/22] * Sqrt[1 + b^2]/b, 10, 100][[1]]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|